Explanation:
1. In order for the idea of "perpendicular distance" to make any sense in this context, the number of sides of the polygon must be even. Then the "diameter" is the diameter of the inscribed circle. As the number of sides increases, the polygon differs less and less from a circle, so the relationship of perimeter and "diameter" becomes closer to the relationship in a circle.
"As the number of sides in a regular polygon increases, the ratio of perimeter to diameter for that polygon approaches pi."
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2. We know from the first question that ...
circumference/diameter = π
And we know that ...
diameter = 2·radius
Then the following are true:
A. circumference = π · diameter
B. circumference = (2·radius) · π
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3. The expressions in order evaluate to approximately ...
4.0, 3.45, 3.31, 3.24
Of these, the last is closest to pi (3.14....). The appropriate choice is ...
D. (10·5)/15.4
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4. Pi cannot be expressed as a rational number, because pi is irrational. A number of lengthy proofs have been offered to demonstrate this fact. One of them makes use of the fact that the tangent of any rational number is irrational, and the tangent of π/4 is 1. Since 1 is a rational number, π/4 cannot be, so π cannot be expressed as a rational number.
